Exploring quasi-geometric frameworks for quantum error-correcting codes: a systematic review

This study investigates quasi-geometric strategies for improving quantum error correction in quantum computing, utilizing geometric principles to improve error detection and correction while maintaining computational efficiency. A comprehensive review of 20 studies, selected from 2988 publications spanning 2019 to 2024, reveals significant progress in quasi-cyclic codes, quasi-orthogonal codes, and quasi-structured geometric codes, highlighting their growing importance in quantum error correction and information theory. The findings indicate that quasi-orthogonal codes that employ coefficient vector differential quasi-orthogonal space-time frequency coding demonstrated a 1.20 dB gain at a bit error rate of 10-4, while reducing computational complexity. Quasi-structured geometric codes offered energy-efficient solutions, facilitating multi-state orthogonal signaling and reliable linear code construction. Furthermore, quasi-cyclic low-density parity-check codes with optimized information selection surpassed traditional forward error correction codes, achieving superior quantum error rates of 10-5 at 10.00 dB and 10-6 at 15.00 dB. Performance analysis showed that the effectiveness of error correction depends more on the frequency of six-length cycles than on girth, suggesting a new direction for optimization. The study emphasizes the transformative potential of quasi-geometric strategies in improving quantum communication by focusing on bit and quantum bit error rates within both stabilizer and classical frameworks. Future work focuses on integrating hybrid quantum-classical codes to raise error resilience and efficiency, addressing challenges like decoding instability, and limited orthogonality to enable reliable and computational quantum communication systems.

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